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Source. Theorem 2.7, p. 5, of Anders Johansson, Jeff Kahn and Van Vu, Factors in random graphs, Random Structures Algorithms 33 (2008), no. 1, 1–28, doi:10.1002/rsa.20224. Labels and pages are those of arXiv:0803.3406v1 (24 March 2008), the edition named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The paper does not write out the proof; Section 12 (pp. 27–28) says the graph proof carries over. Nothing here is independently reviewed.

Statement

Setting. $H_k(n,p)$, $\mathrm{th}_H(n)$ and $d^*(H)$, the maximum of $d(H')=e(H')/(v(H')-1)$ over subhypergraphs $H'\subseteq H$, are as on the pages for Theorem 2.5 and Theorem 2.2; the paper says the graph definitions extend without modification (p. 5).

Theorem 2.7 (p. 5). For an arbitrary $k$-uniform hypergraph $H$,

$$\mathrm{th}_H(n)=O\bigl(n^{-1/d^*(H)+o(1)}\bigr).$$

The paper says (p. 6) that its counting version also holds.

Proof pointer

Not written out. Section 12 (pp. 27–28) combines the two modifications it describes: taking $p\ge n^{-1/d^*(H)+\epsilon}$, as for Theorem 2.2, and noting that the arguments make no use of edges having size two.

Dependencies

None in the corpus. Internal: the proof of Theorem 2.4, transferred as Section 12 describes.

Bears on

No Erdős problem.